Optimal. Leaf size=138 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{7/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{b^2 \log (x)}{9 a^{7/3}}+\frac{b \left (a+b x^2\right )^{2/3}}{3 a^2 x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 a x^4} \]
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Rubi [A] time = 0.228244, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{7/3}}+\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{b^2 \log (x)}{9 a^{7/3}}+\frac{b \left (a+b x^2\right )^{2/3}}{3 a^2 x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(a + b*x^2)^(1/3)),x]
[Out]
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Rubi in Sympy [A] time = 15.961, size = 126, normalized size = 0.91 \[ - \frac{\left (a + b x^{2}\right )^{\frac{2}{3}}}{4 a x^{4}} + \frac{b \left (a + b x^{2}\right )^{\frac{2}{3}}}{3 a^{2} x^{2}} - \frac{b^{2} \log{\left (x^{2} \right )}}{18 a^{\frac{7}{3}}} + \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{6 a^{\frac{7}{3}}} + \frac{\sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(b*x**2+a)**(1/3),x)
[Out]
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Mathematica [C] time = 0.0539015, size = 82, normalized size = 0.59 \[ \frac{-3 a^2-4 b^2 x^4 \sqrt [3]{\frac{a}{b x^2}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^2}\right )+a b x^2+4 b^2 x^4}{12 a^2 x^4 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(a + b*x^2)^(1/3)),x]
[Out]
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Maple [F] time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(b*x^2+a)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222643, size = 196, normalized size = 1.42 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + a\right ) - 4 \, \sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} - a\right ) - 12 \, b^{2} x^{4} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (4 \, b x^{2} - 3 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{1}{3}}\right )}}{108 \, a^{\frac{7}{3}} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.55842, size = 41, normalized size = 0.3 \[ - \frac{\Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 \sqrt [3]{b} x^{\frac{14}{3}} \Gamma \left (\frac{10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(b*x**2+a)**(1/3),x)
[Out]
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GIAC/XCAS [A] time = 0.598851, size = 171, normalized size = 1.24 \[ \frac{1}{36} \, b^{2}{\left (\frac{4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{7}{3}}} - \frac{2 \,{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{7}{3}}} + \frac{4 \,{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{7}{3}}} + \frac{3 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{a^{2} b^{2} x^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/3)*x^5),x, algorithm="giac")
[Out]